The rank part is completely open when \(\mathrm {rk}_{\mathbb {Z}} E(\mathbb {Q})\)  or \(\mathrm {ord}_{s=1} L(E, s)\)  is larger than 1.
Thanks to the work of Coates–Wiles [1]}, Rubin [2]}, Gross–Zagier [3]}, Kolyvagin [4]}, and Kato [5]} based on the method of Euler systems, it is known that
if \(\mathrm {ord}_{s=1}L(E,s) \le 1\) , then the rank part is true and the Tate–Shafarevich group is finite.
Also, its \(p\) -converse is recently developed enormously; see [6]}, [7]}, [8]}, [9]}, [10]} for example.
Furthermore, when \(\mathrm {ord}_{s=1}L(E,s) \le 1\) , a large amount of the \(p\) -part of the BSD formula (REF ) is resolved through the establishment of various Iwasawa main conjectures and complex and \(p\) -adic Gross–Zagier formulas
thanks to the work of Skinner–Urban [6]}, Kobayashi [12]}, Zhang [8]}, Wan [14]}, [15]}, Berti–Bertolini–Venerucci [16]}, Jetchev–Skinner–Wan [17]}, Castella [18]}, Castella–Grossi–Lee–Skinner [19]}, and Büyükboduk–Pollack–Sasaki [20]}.
